Continuity Defined

CONTINUITY

For a function to be continuous, the rule of thumb is that you can draw the graph of this continuous function with a continuous line. That is, you don't have to pick up your pen as you draw the curve.

More formally, a function is discontinuous at vertical asymptotes and other values of x that don't belong to the domain of the function. Points where no y-value can be defined.

For example if , we know that f (x) is continuous on its domain of . This function is discontinuous when x > 5 since, in the set of Reals, we cannot find an even root for a negative number. So, f (x) is discontinuous on intervals where (x, f (x)) are not defined by the rule of correspondence.

Generally, when asked to find where a function is continuous, we find where it is discontinuous and say that it's continuous everywhere else.

The Test for Continuity

There are 3 parts to the test for continuity.

To test if a function f(x) is continuous at the point x = a, test if

1) f (a) is defined.

2) exists.

3) .

What does each item in the list mean?

1) f (a) is defined means that we get a y-value when we substitute x = a.

(a, f (a))

2) for to exist, must equal .

Limit from the left must equal limt from the right.

3) f (a) = means the y-value of part 1 must equal the limit of part 2.

continuity intro test for continuity examples

practice solutions

For example:

Test the function for continuity at x = 1.

test part 1

f (1) = 2(1) – 1 = 1, so f (1) is defined.

test part 2

, but .

Since 1 doesn't = 2, f (x) is discontinuous at x = 1.

f (x) fails part 2 of the test for continuity.

Had the function been ,

it would be continuous at x = 1 since

both the left and right side limits would = 1 which equals f (1).

This function would pass all 3 parts of the continuity test.

Example: On the curve below, test the continuity at x = 3:

f (3) = 2 so part 1 works.

, so part 2 works.

But 2 doesn't = 6, so part 3 doesn't work.
The y-value at "a" must equal the limit value at "a".

If we set f (3) = 6, then this function is continuous at x = 3.

Classic Exam Question:

Given , find a value for c that makes f (x) continuous at x = 2.

First, we'll find f (2), then we'll find the limit as x approaches 2, and then we'll set them equal to satisfy the statements of the test for continuity.

f (2) = 4c – 3

now solve left limit = right limit:

4c – 3 = 2c + 2 gives us c = 5/2 or 2.5.

This makes f (2) = 7 and limit as x approaches 2 = 7.

In similar questions with 2 unknowns -- say c and d, we get enough information to find a system of 2 equations in order to solve for the 2 unknowns. In such cases, there will also be 2 different values to limit towards. See practice question 2b.

continuity intro test for continuity examples

practice solutions

Practice

1) Discuss the continuity of these functions:

a) b) c)

2) Determine which values of c and/or d make f (x) continuous on all Real numbers.

a) Given

b) Given

continuity intro test for continuity examples

practice solutions

Solutions

1) Discuss the continuity of these functions:

a)
f (x) continuous on
b)
g(x) continuous on
(vertical asymptotes)

c) , here we have 3 situations to discuss.
1/ The domain of
2/ The domain of
3/ The restriction on (x – 4) in the denominator is
h (x) continuous on [ – 5, – 3] [3, 4) (4, 5]
interval notation:

2) Determine which values of c and/or d make f (x) continuous on all Real numbers.

a) Given

First, we'll find f(1), then we'll find the limit as x approaches 1, and then we'll set them equal to satisfy the statements of the test for continuity.

f (1) = 3c – 2